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1 Methods

Validation of assay methods is based on the use of calibration and validation standards made by accurately weighing predetermined quantities of analytes as required. The experiments are generally carried out on multiple series (usually carried out on different days) on multiple concentration levels with multiple replicates per level and per series.

In order to complete this report, it is recommended to annex the exact drug formula with the reference of each of the raw materials used as well as a complete description of pre-analytical and analytical methods used.

1.1 Calibration curves

The estimation of the parameters of the calibration curves were obtained from the ordinary least squares method (OLS) using the r stats::lm() function.

When possible various linear regressions were tested:

  • Linear function (Linear)
  • Linear trough 0 function (Linear 0)
  • Linear trough 0 and the highest concentration levels function (Linear 0 - Max)
  • Linear function with weights (1/Y and 1/X) (Linear weighed 1/X and Linear weighed 1/Y)

1.2 Estimation of trueness and precision

1.2.1 Models

Considering \(i \in [1;p]\) (series index), \(j \in [1,m]\) (concentration index), \(k\in [1;n]\) (repetition index) and \(x_{i,j,k, calc}\) the \(k\)th back calculated concentration of the \(i\) series \(j\) level (using calibration standards) :

\[ x_{i,j,k, calc} = \mu_j + \alpha_{i,j} + \epsilon_{i,j,k} \]

With : \(\begin{cases} \mu_j &: \text{the back calculated mean of the $j$ level} \\ \alpha_{i,j} &: \text{at $j$ level, the difference between the $i$th series average and the $\mu_j$}\\ \epsilon_{i,j,k} &: \text{the experimental error} \end{cases}\)

\(\begin{cases}\alpha_{i,j} &\sim \mathcal{N}(0,\sigma^2_{B,j}) \; \text{with $\sigma^2_{B,j}$ representing the interseries variances} \\ \epsilon_{i,j,k} &\sim \mathcal{N}(0, \sigma^2_{W,j}) \; \text{with $\sigma^2_{W,j}$ representing the intraseries variances}\end{cases}\)

1.2.2 Trueness

Trueness expresses the closeness of agreement between the mean value obtained from a series of test results and an accepted reference value. The trueness gives an indication of the systemic errors. It was calculated at each concentration level by calculating the difference between the introduced concentrations (arithmetic) mean \(\overline{x}_j\) and the calculated concentrations mean \(\hat{\mu}_j\).

The bias (for j th level) was expressed as: \(\text{Bias}_j =\hat{\mu}_j-\overline{x}_j\)

And the relative bias as: \(\text{Bias(%)}_j = \frac{\hat{\mu}_j-\overline{x}_j}{\overline{x}_j}\times 100\)

The recovery (for j th level) was expressed as : \(\text{Recovery(%)}_j = \frac{\hat{\mu}_j}{\overline{x}_j}\times 100\)

1.2.3 Fidelity & intermediate precision

Fidelity expresses the closeness of agreement between a series of measurements from multiple takes of the same homogeneous sample, under prescribed conditions. It provides information on random errors and is assessed at two levels, repeatability and intermediate precision.

Intra and interseries variance could be estimated at every \(j\) level using the restricted maximum likelihood method:

\(\begin{align} \hat{\mu}_j &= \frac{1}{\sum^p_{i=1}n_{i,j}}\cdot\sum^p_{i=1}\sum^{n_{i,j}}_{k=1} x_{i,j,k, calc} \\ \text{MSM}_j &= \frac{1}{p-1} \cdot \sum^p_{i=1}n_{i,j}\cdot \left(x_{i,j, calc}-\overline{x}_{i,j, calc}\right)^2 \\ \text{MSE}_j &=\frac{1}{\sum^p_{i=1}n_{i,j}-p}\cdot\sum^p_{i=1}\sum^{n_{i,j}}_{k=1} \left(x_{i,j, calc}-\overline{x}_{i,j, calc}\right)^2 \end{align}\)

\(\text{if MSE$_j$ < MSM$_j$} \begin{cases} \hat{\sigma}^2_{W,j} &=\text{MSE}_j \\ \hat{\sigma}^2_{B,j} &= \frac{\text{MSM}_j-\text{MSE}_j}{n} \end{cases}\)

\(\text{else} \begin{cases} \hat{\sigma}^2_{W,j} &=\frac{1}{pn - 1}\cdot\sum^p_{i=1}\sum^{k}_{j=1} \left(x_{i,j,k calc}-\overline{x}_{j, calc}\right)^2 \\ \hat{\sigma}^2_{B,j} &= 0 \end{cases}\)

Intermediate precision was then calculated: \(\hat{\sigma}^2_{IP,j} = \hat{\sigma}^2_{W,j} + \hat{\sigma}^2_{B,j}\)

Each corresponding coefficient of variation was determined as: \(\text{CV(%)} = \frac{\sigma_j}{\hat{\mu}_j}\)

1.2.4 Tolerance interval and accuracy profiles

The tolerance interval is a statistical interval within which, with some confidence level, a specified proportion (\(\beta\)) of a sampled population falls. The construction of the tolerance intervals using standard solutions therefore makes it possible to predict with a some confidence levels where a proportion of the dosages will falls.

The tolerance interval has been computed, at each concentration levels using validation standards as follows:

  • \(\hat{\mu}_j \pm \mathcal{Q}t(\upsilon,\frac{1 + \beta}{2})\cdot\sqrt{1 + \frac{1}{pn\cdot B^2_j}}\cdot\hat\sigma_{IP,j}\)

  • \(\text{Bias(%)} \pm \mathcal{Q}_t(\upsilon,\frac{1 + \beta}{2}) \cdot \sqrt{1 + \frac{1}{pn\cdot B^2_j}} \cdot \hat{ \text{CV}}_{IP,j}\)

\(with \begin{cases}\begin{align} R_j &= \frac{\sigma^2_{B,j}}{\sigma^2_{W,j}} \\ B_j &= \sqrt{\frac{R_j +1}{n\cdot R_j + 1}} \\ \upsilon &= \frac{(R+1)^2}{\frac{(R + (1/n))^2}{p - 1} + \frac{(1 - 1/n)}{pn} } \end{align}\end{cases}\)

\(\mathcal{Q}_t(\upsilon,\frac{1 + \beta}{2})\) corresponded to the \(\beta\) quantile of the Student \(t\) distribution with \(\upsilon\) degrees of freedom.

  • The \(\beta\) value used for the calculation of the \(\beta\)-expectation tolerance interval was set to: 0.8 (user defined).

The accuracy profiles (─) were plotted joining the tolerance intervals obtained for each of the levels tested. A method is validated over the full range of measurements, when the accuracy profile (─) is fully included between the upper and lower bound of the acceptability limit \([-\lambda;+\lambda]\) (┄) set a priori. The method can only be used over the concentration range for which the accuracy profile is entirely within the acceptability limit \([-\lambda;+\lambda]\).

  • The acceptability value used was set to: 10 % (user defined).

1.2.5 Determination of quantification limits.

When the accuracy profile (─) is not fully included within the acceptability limit \([-\lambda;+\lambda]\) ( ┄), a limit of quantification can be determined at the intersection of the accuracy profile with the upper (\(+\lambda\)) or lower (\(-\lambda\)) acceptability limit. This can happen at low or high concentrations and can produce lower or higher LOQs respectively.

The coordinates of the intersections between the accuracy profiles and the acceptability limits were calculated and plotted on the tolerance profiles (✴)

  • The lower LOQ is the lowest amount of analyte in a sample which can be quantitatively determined with suitable precision and accuracy. It should be determined at lower levels using the last intersection point between accuracy profile and acceptance limits, before accuracy profile become fully included in acceptance limits. It’s the first level tested if accuracy profile is fully inside acceptance limits at this level.

  • The upper LOQ is the highest amount of analyte in a sample which can be quantitatively determined with suitable precision and accuracy. It should be determined at higher levels using the first intersection point between accuracy profile and acceptance limits, before accuracy profile partly or fully exceeds acceptance limits. It is the last level tested if accuracy profile is fully inside acceptance limits from the lower LOQ to this level.

2 Analysis of response function (calibration curves)

2.1 Methods and Data

Response functions \(signal=f(concentration)\) were analysed using stats4::lm() function in R, using calibration data provided in Table 4.1 and Figure 4.1) (see section 4).

Analysis was performed using :

  • Linear function (Linear)
  • Linear trough 0 function (Linear 0)
  • Linear trough 0 and the highest concentration levels function (Linear 0 - Max)
  • Linear function with weights (1/Y and 1/X) (Linear weighed 1/X and Linear weighed 1/Y)

2.2 Response functions obtained

2.2.1 Regression analysis

The interactive table 2.1 shows the values obtained with regressions :

Table 2.1: Results of linear regressions performed
Methods Serie Intercept Slope AIC R²
Linear (LM) 1 -1358.7877 7566838 188.82847 0.9987755
Linear (LM) 2 -2972.4758 7166164 157.35429 0.9999733
Linear (LM) 3 -4450.9923 7101554 178.31360 0.9996262
Linear 0 (LM) 1 0.0000 7559372 186.84928 0.9991447
Linear 0 (LM) 2 0.0000 7149831 159.29751 0.9999694
Linear 0 (LM) 3 0.0000 7077097 177.10514 0.9997112
Linear 0-max (LM) 1 0.0000 7559800 52.44744 0.9991522
Linear 0-max (LM) 2 0.0000 7151585 43.89910 0.9999868
Linear 0-max (LM) 3 0.0000 7083672 48.34795 0.9998756
Linear weighed 1/X (LM) 1 -1461.3560 7568686 165.82140 0.9987859
Linear weighed 1/X (LM) 2 -2165.6005 7151626 156.85434 0.9995574
Linear weighed 1/X (LM) 3 -530.7419 7030919 166.86941 0.9984005
Linear weighed 1/Y (LM) 1 -1687.6258 7562968 165.13549 0.9986793
Linear weighed 1/Y (LM) 2 -2215.1444 7146698 159.77107 0.9991691
Linear weighed 1/Y (LM) 3 -575.2321 7018751 167.75477 0.9981079

2.2.2 Residues for each calibration curve at each level

The residues obtained are shown in the interactive figure 2.1.

Figure 2.1: Relative bias calculated from regression

3 Validation

3.1 Using a linear calibration curve

3.1.1 Trueness and precision obtained

Trueness and precision are depicted on table 3.1 and 3.2

Table 3.1: Trueness and precision estimators and limits ( LIN )
Introduced concentrations default Mean calculated concentrations default Bias default Repeatability SD default Between series SD default Intermediate precision SD default Low limit of tolerance default High limit of tolerance default
0.0005 0.0007239 0.0002239 0.0001373 1.0e-07 0.0002666 0.0002355 0.0012122
0.0015 0.0016881 0.0001881 0.0002630 0.0e+00 0.0003053 0.0012286 0.0021476
0.0200 0.0195866 -0.0004134 0.0014254 6.0e-07 0.0016220 0.0171659 0.0220074
0.2000 0.1999305 -0.0000695 0.0114414 7.6e-06 0.0117702 0.1830347 0.2168264
Table 3.2: Trueness and precision estimators and limits ( LIN )
Introduced concentrations default Mean calculated concentrations default Bias (%) Recovery (%) CV repeatability (%) CV intermediate precision (%) Low limit of tolerance (%) High limit of tolerance (%) Results
0.0005 0.0007239 44.775711 144.77571 27.459330 53.325616 -52.893662 142.445083 FAIL
0.0015 0.0016881 12.540253 112.54025 17.534663 20.355825 -18.092530 43.173037 FAIL
0.0200 0.0195866 -2.066788 97.93321 7.126778 8.109942 -14.170430 10.036855 FAIL
0.2000 0.1999305 -0.034729 99.96527 5.720705 5.885101 -8.482649 8.413191 PASS

3.1.2 Accuracy profile

Accuracy profile is shown Figure 3.1. The 𝛽-tolerance interval (─) should be entirely within the acceptance limits (┄). Coordinates of points of intersection between the tolerance interval and the acceptability limit were marked with a red star (✴) when present.

Figure 3.1: Accuracy profiles (red dashed line: acceptance limits, blue lines: 𝛽-tolerance intervals).

3.1.3 Linearity Profile

Linearity profile is shown Figure 3.2. You should check that the black line (linear model) should be superimposed on the red dashed identity line.

Figure 3.2: Linearity profiles (red dashed line: identity line, black lines: linear regression lines, blue lines: 𝛽-tolerance intervals).

3.1.4 Linear regression

The p-value for the intercept is > 0.05 which is in favor of a non-significant value (PASS)

The p-value for the slope is < 0.05 which is in favor of a significant value (PASS)

Table 3.3: Results of the linear regression.
Estimate CI (lower) CI (upper) Std. Error t value Pr(>|t|)
(Intercept) 0.0000175 -0.0019941 0.0020292 0.0009994 0.0175485 0.986
x 0.9993649 0.9793491 1.0193808 0.0099438 100.5013829 <0.001 ***

3.1.5 Studentized Breusch-Pagan test for heteroskedasticity

The p-value < 0.05 which is in favor of heterosedasticity and standard deviations of a calculated concentrations as related to introduced concentrations, are non-constant (this should be investigated)

Result of the studentized Breusch-Pagan test
Test statistic df P value
5.044 1 0.02471 *

3.2 Using a linear forced through 0 calibration curve

3.2.1 Trueness and precision obtained

Trueness and precision are depicted on the table 3.4 and 3.5

Table 3.4: Trueness and precision estimators and limits ( LIN_0 )
Introduced concentrations default Mean calculated concentrations default Bias default Repeatability SD default Between series SD default Intermediate precision SD default Low limit of tolerance default High limit of tolerance default
0.0005 0.0003176 -0.0001824 0.0001376 0.0e+00 0.0001376 0.0001219 0.0005133
0.0015 0.0012839 -0.0002161 0.0002636 0.0e+00 0.0002636 0.0009091 0.0016587
0.0200 0.0192220 -0.0007780 0.0014294 8.0e-07 0.0016996 0.0166384 0.0218056
0.2000 0.1999679 -0.0000321 0.0114542 7.5e-06 0.0117785 0.1830626 0.2168732
Table 3.5: Trueness and precision estimators and limits ( LIN_0 )
Introduced concentrations default Mean calculated concentrations default Bias (%) Recovery (%) CV repeatability (%) CV intermediate precision (%) Low limit of tolerance (%) High limit of tolerance (%) Results
0.0005 0.0003176 -36.4882832 63.51172 27.525332 27.525332 -75.627610 2.651044 FAIL
0.0015 0.0012839 -14.4058986 85.59410 17.570893 17.570893 -39.390627 10.578829 FAIL
0.0200 0.0192220 -3.8899655 96.11003 7.146753 8.497997 -16.808072 9.028141 FAIL
0.2000 0.1999679 -0.0160614 99.98394 5.727122 5.889251 -8.468712 8.436589 PASS

3.2.2 Accuracy profile

Accuracy profile is shown in Figure 3.3. The 𝛽-tolerance interval (─) should be entirely within the acceptance limits (┄). Coordinates of points of intersection between the tolerance interval and the acceptability limit were marked with a red star (✴) when present.

Figure 3.3: Accuracy profiles (red dashed line: acceptance limits, blue lines: 𝛽-tolerance intervals).

3.2.3 Linearity Profile

Linearity profile is shown in Figure 3.4. You should check that the black line (linear model) should be superimposed on the red dashed identity line.

Figure 3.4: Linearity profiles (red dashed line: identity line, black lines: linear regression lines, blue lines: 𝛽-tolerance intervals).

3.2.4 Linear regression

The p-value for the intercept is > 0.05 which is in favor of a non-significant value (PASS)

The p-value for the slope is < 0.05 which is in favor of a significant value (PASS)

Table 3.6: Results of the linear regression.
Estimate CI (lower) CI (upper) Std. Error t value Pr(>|t|)
(Intercept) -0.0003905 -0.0024046 0.0016237 0.0010006 -0.3902287 0.698
x 1.0015911 0.9815504 1.0216317 0.0099561 100.6004122 <0.001 ***

3.2.5 Studentized Breusch-Pagan test for heteroskedasticity

The p-value < 0.05 which is in favor of heterosedasticity and standard deviations of a calculated concentrations as related to introduced concentrations, are non-constant (this should be investigated)

Result of the studentized Breusch-Pagan test
Test statistic df P value
5.05 1 0.02462 *

3.3 Using a linear calibration curve with the highest level only and forced through 0

3.3.1 Trueness and precision obtained

Trueness and precision are depicted on the table 3.7 and 3.8

Table 3.7: Trueness and precision estimators and limits ( LIN_0MAX )
Introduced concentrations default Mean calculated concentrations default Bias default Repeatability SD default Between series SD default Intermediate precision SD default Low limit of tolerance default High limit of tolerance default
0.0005 0.0003174 -0.0001826 0.0001376 0.0e+00 0.0001376 0.0001218 0.0005130
0.0015 0.0012834 -0.0002166 0.0002635 0.0e+00 0.0002635 0.0009088 0.0016580
0.0200 0.0192142 -0.0007858 0.0014286 9.0e-07 0.0017009 0.0166273 0.0218012
0.2000 0.1998863 -0.0001137 0.0114533 7.8e-06 0.0117893 0.1829597 0.2168130
Table 3.8: Trueness and precision estimators and limits ( LIN_0MAX )
Introduced concentrations default Mean calculated concentrations default Bias (%) Recovery (%) CV repeatability (%) CV intermediate precision (%) Low limit of tolerance (%) High limit of tolerance (%) Results
0.0005 0.0003174 -36.5154449 63.48456 27.511253 27.511253 -75.634752 2.603862 FAIL
0.0015 0.0012834 -14.4401297 85.55987 17.563399 17.563399 -39.414202 10.533942 FAIL
0.0200 0.0192142 -3.9288280 96.07117 7.142830 8.504340 -16.863670 9.006014 FAIL
0.2000 0.1998863 -0.0568324 99.94317 5.726666 5.894633 -8.520142 8.406478 PASS

3.3.2 Accuracy profile

Accuracy profile is shown in Figure 3.5. The 𝛽-tolerance interval (─) should be entirely within the acceptance limits (┄). Coordinates of points of intersection between the tolerance interval and the acceptability limit were marked with a red star (✴) when present.

Figure 3.5: Accuracy profiles (red dashed line: acceptance limits, blue lines: 𝛽-tolerance intervals).

3.3.3 Linearity Profile

Linearity profile is shown in Figure 3.6. You should check that the black line (linear model) should be superimposed on the red dashed identity line.

Figure 3.6: Linearity profiles (red dashed line: identity line, black lines: linear regression lines, blue lines: 𝛽-tolerance intervals).

3.3.4 Linear regression

The p-value for the intercept is > 0.05 which is in favor of a non-significant value (PASS)

The p-value for the slope is < 0.05 which is in favor of a significant value (PASS)

Table 3.9: Results of the linear regression.
Estimate CI (lower) CI (upper) Std. Error t value Pr(>|t|)
(Intercept) -0.0003903 -0.0024057 0.0016251 0.0010013 -0.3897947 0.698
x 1.0011826 0.9811289 1.0212362 0.0099626 100.4942902 <0.001 ***

3.3.5 Studentized Breusch-Pagan test for heteroskedasticity

The p-value < 0.05 which is in favor of heterosedasticity and standard deviations of a calculated concentrations as related to introduced concentrations, are non-constant (this should be investigated)

Result of the studentized Breusch-Pagan test
Test statistic df P value
5.025 1 0.02498 *

3.4 Using a weighed (1/Y) linear calibration curve

3.4.1 Trueness and precision obtained

Trueness and precision are depicted on the table 3.10 and 3.11

Table 3.10: Trueness and precision estimators and limits ( LIN_1Y )
Introduced concentrations default Mean calculated concentrations default Bias default Repeatability SD default Between series SD default Intermediate precision SD default Low limit of tolerance default High limit of tolerance default
0.0005 0.0005235 0.0000235 0.0001382 0.0e+00 0.0001529 0.0002980 0.0007490
0.0015 0.0014923 -0.0000077 0.0002644 0.0e+00 0.0002845 0.0010772 0.0019074
0.0200 0.0194793 -0.0005207 0.0014350 7.0e-07 0.0016692 0.0169653 0.0219933
0.2000 0.2007220 0.0007220 0.0114504 6.3e-06 0.0117212 0.1839255 0.2175185
Table 3.11: Trueness and precision estimators and limits ( LIN_1Y )
Introduced concentrations default Mean calculated concentrations default Bias (%) Recovery (%) CV repeatability (%) CV intermediate precision (%) Low limit of tolerance (%) High limit of tolerance (%) Results
0.0005 0.0005235 4.7026406 104.70264 27.634133 30.571414 -40.402985 49.808267 FAIL
0.0015 0.0014923 -0.5119997 99.48800 17.627489 18.963577 -28.185581 27.161582 FAIL
0.0200 0.0194793 -2.6034781 97.39652 7.174993 8.346037 -15.173646 9.966690 FAIL
0.2000 0.2007220 0.3609959 100.36100 5.725221 5.860588 -8.037274 8.759265 PASS

3.4.2 Accuracy profile

Accuracy profile is shown in Figure 3.7. The 𝛽-tolerance interval (─) should be entirely within the acceptance limits (┄). Coordinates of points of intersection between the tolerance interval and the acceptability limit were marked with a red star (✴) when present.

Figure 3.7: Accuracy profiles (red dashed line: acceptance limits, blue lines: 𝛽-tolerance intervals).

3.4.3 Linearity Profile

Linearity profile is shown in Figure 3.8. You should check that the black line (linear model) should be superimposed on the red dashed identity line.

Figure 3.8: Linearity profiles (red dashed line: identity line, black lines: linear regression lines, blue lines: 𝛽-tolerance intervals).

3.4.4 Linear regression

The p-value for the intercept is > 0.05 which is in favor of a non-significant value (PASS)

The p-value for the slope is < 0.05 which is in favor of a significant value (PASS)

Table 3.12: Results of the linear regression.
Estimate CI (lower) CI (upper) Std. Error t value Pr(>|t|)
(Intercept) -0.0001867 -0.0021933 0.0018199 0.0009969 -0.187296 0.852
x 1.0043421 0.9843767 1.0243076 0.0099188 101.256809 <0.001 ***

3.4.5 Studentized Breusch-Pagan test for heteroskedasticity

The p-value < 0.05 which is in favor of heterosedasticity and standard deviations of a calculated concentrations as related to introduced concentrations, are non-constant (this should be investigated)

Result of the studentized Breusch-Pagan test
Test statistic df P value
5.346 1 0.02078 *

3.5 Using a weighed (1/X) linear calibration curve

3.5.1 Trueness and precision obtained

Trueness and precision are depicted on the table 3.13 and 3.14

Table 3.13: Trueness and precision estimators and limits ( LIN_1X )
Introduced concentrations default Mean calculated concentrations default Bias default Repeatability SD default Between series SD default Intermediate precision SD default Low limit of tolerance default High limit of tolerance default
0.0005 0.0005086 0.0000086 0.0001380 0.0e+00 0.0001526 0.0002835 0.0007337
0.0015 0.0014764 -0.0000236 0.0002641 0.0e+00 0.0002842 0.0010615 0.0018913
0.0200 0.0194443 -0.0005557 0.0014333 7.0e-07 0.0016599 0.0169489 0.0219398
0.2000 0.2004948 0.0004948 0.0114418 5.9e-06 0.0116970 0.1837403 0.2172492
Table 3.14: Trueness and precision estimators and limits ( LIN_1X )
Introduced concentrations default Mean calculated concentrations default Bias (%) Recovery (%) CV repeatability (%) CV intermediate precision (%) Low limit of tolerance (%) High limit of tolerance (%) Results
0.0005 0.0005086 1.7219357 101.72194 27.600381 30.520939 -43.301462 46.745333 FAIL
0.0015 0.0014764 -1.5721901 98.42781 17.607017 18.949962 -29.230626 26.086246 FAIL
0.0200 0.0194443 -2.7783341 97.22167 7.166691 8.299697 -15.255678 9.699010 FAIL
0.2000 0.2004948 0.2473866 100.24739 5.720891 5.848517 -8.129828 8.624601 PASS

3.5.2 Accuracy profile

Accuracy profile is shown in Figure 3.9. The 𝛽-tolerance interval (─) should be entirely within the acceptance limits (┄). Coordinates of points of intersection between the tolerance interval and the acceptability limit were marked with a red star (✴) when present.

Figure 3.9: Accuracy profiles (red dashed line: acceptance limits, blue lines: 𝛽-tolerance intervals).

3.5.3 Linearity Profile

Linearity profile is shown in Figure 3.10. You should check that the black line (linear model) should be superimposed on the red dashed identity line.

Figure 3.10: Linearity profiles (red dashed line: identity line, black lines: linear regression lines, blue lines: 𝛽-tolerance intervals).

3.5.4 Linear regression

The p-value for the intercept is > 0.05 which is in favor of a non-significant value (PASS)

The p-value for the slope is < 0.05 which is in favor of a significant value (PASS)

Table 3.15: Results of the linear regression.
Estimate CI (lower) CI (upper) Std. Error t value Pr(>|t|)
(Intercept) -0.0002008 -0.0022039 0.0018022 0.0009951 -0.2018264 0.841
x 1.0032769 0.9833468 1.0232071 0.0099012 101.3285576 <0.001 ***

3.5.5 Studentized Breusch-Pagan test for heteroskedasticity

The p-value < 0.05 which is in favor of heterosedasticity and standard deviations of a calculated concentrations as related to introduced concentrations, are non-constant (this should be investigated)

Result of the studentized Breusch-Pagan test
Test statistic df P value
5.3 1 0.02132 *

4 Raw data

User provided.

Table 4.1: Calibration standards raw data
ID TYPE SERIE SIGNAL CONC_LEVEL
CAL CAL 1 3057 0.00049
CAL CAL 1 11600 0.00150
CAL CAL 1 155620 0.02000
CAL CAL 1 1556003 0.20000
CAL CAL 1 1720 0.00050
CAL CAL 1 7430 0.00150
CAL CAL 1 145383 0.02000
CAL CAL 1 1467917 0.20000
CAL CAL 2 2144 0.00050
CAL CAL 2 5250 0.00150
CAL CAL 2 142242 0.02000
CAL CAL 2 1425120 0.20000
CAL CAL 2 1847 0.00050
CAL CAL 2 8487 0.00150
CAL CAL 2 137393 0.02000
CAL CAL 2 1435514 0.20000
CAL CAL 3 2744 0.00050
CAL CAL 3 10473 0.00150
CAL CAL 3 141074 0.02000
CAL CAL 3 1400930 0.20000
CAL CAL 3 3422 0.00050
CAL CAL 3 10554 0.00150
CAL CAL 3 115746 0.02000
CAL CAL 3 1432539 0.20000

Figure 4.1: Calibration data

Table 4.2: Validation standards raw data
ID TYPE SERIE SIGNAL CONC_LEVEL
VAL VAL 1 1544 0.0005
VAL VAL 1 9992 0.0015
VAL VAL 1 155683 0.0200
VAL VAL 1 1767753 0.2000
VAL VAL 1 1422 0.0005
VAL VAL 1 11212 0.0015
VAL VAL 1 154877 0.0200
VAL VAL 1 1540288 0.2000
VAL VAL 1 3661 0.0005
VAL VAL 1 11527 0.0015
VAL VAL 1 154062 0.0200
VAL VAL 1 1466979 0.2000
VAL VAL 1 2712 0.0005
VAL VAL 1 6007 0.0015
VAL VAL 1 153763 0.0200
VAL VAL 1 1466715 0.2000
VAL VAL 2 2593 0.0005
VAL VAL 2 10558 0.0015
VAL VAL 2 140350 0.0200
VAL VAL 2 1413294 0.2000
VAL VAL 2 2880 0.0005
VAL VAL 2 9174 0.0015
VAL VAL 2 110853 0.0200
VAL VAL 2 1425878 0.2000
VAL VAL 2 1316 0.0005
VAL VAL 2 9065 0.0015
VAL VAL 2 136750 0.0200
VAL VAL 2 1368431 0.2000
VAL VAL 2 1908 0.0005
VAL VAL 2 9657 0.0015
VAL VAL 2 130685 0.0200
VAL VAL 2 1332873 0.2000
VAL VAL 3 1498 0.0005
VAL VAL 3 7730 0.0015
VAL VAL 3 142770 0.0200
VAL VAL 3 1430404 0.2000
VAL VAL 3 3415 0.0005
VAL VAL 3 10179 0.0015
VAL VAL 3 141127 0.0200
VAL VAL 3 1409384 0.2000
VAL VAL 3 3423 0.0005
VAL VAL 3 10600 0.0015
VAL VAL 3 138551 0.0200
VAL VAL 3 1403500 0.2000
VAL VAL 3 1281 0.0005
VAL VAL 3 6198 0.0015
VAL VAL 3 117689 0.0200
VAL VAL 3 1411380 0.2000

5 R Packages used

Aphalo, Pedro J. 2022a. Ggpmisc: Miscellaneous Extensions to Ggplot2. https://CRAN.R-project.org/package=ggpmisc

Aphalo, Pedro J. 2022b. Ggpp: Grammar Extensions to Ggplot2. https://CRAN.R-project.org/package=ggpp

Dahl, David B., David Scott, Charles Roosen, Arni Magnusson, and Jonathan Swinton. 2019. Xtable: Export Tables to LaTeX or HTML. http://xtable.r-forge.r-project.org/

DarĂłczi, Gergely, and Roman Tsegelskyi. 2022. Pander: An r Pandoc Writer. https://rapporter.github.io/pander/

Dowle, Matt, and Arun Srinivasan. 2022. Data.table: Extension of ‘Data.frame’. https://CRAN.R-project.org/package=data.table

Fox, John, and Sanford Weisberg. 2019. An R Companion to Applied Regression. Third. Thousand Oaks CA: Sage. https://socialsciences.mcmaster.ca/jfox/Books/Companion/

Fox, John, Sanford Weisberg, and Brad Price. 2022a. Car: Companion to Applied Regression. https://CRAN.R-project.org/package=car

Fox, John, Sanford Weisberg, and Brad Price. 2022b. carData: Companion to Applied Regression Data Sets. https://CRAN.R-project.org/package=carData

Garnier, Simon. 2021. Viridis: Colorblind-Friendly Color Maps for r. https://CRAN.R-project.org/package=viridis

Garnier, Simon.. 2022. viridisLite: Colorblind-Friendly Color Maps (Lite Version). https://CRAN.R-project.org/package=viridisLite

Hofner, Benjamin. 2021. papeR: A Toolbox for Writing Pretty Papers and Reports. https://CRAN.R-project.org/package=papeR

Hofner, Benjamin, and with contributions by many others. 2021. papeR: A Toolbox for Writing Pretty Papers and Reports. https://github.com/hofnerb/papeR

Hothorn, Torsten, Achim Zeileis, Richard W. Farebrother, and Clint Cummins. 2022. Lmtest: Testing Linear Regression Models. https://CRAN.R-project.org/package=lmtest

R Core Team. 2022. R: A Language and Environment for Statistical Computing. Vienna, Austria: R Foundation for Statistical Computing. https://www.R-project.org/

Sievert, Carson. 2020. Interactive Web-Based Data Visualization with r, Plotly, and Shiny. Chapman; Hall/CRC. https://plotly-r.com

Sievert, Carson, Chris Parmer, Toby Hocking, Scott Chamberlain, Karthik Ram, Marianne Corvellec, and Pedro Despouy. 2022. Plotly: Create Interactive Web Graphics via Plotly.js. https://CRAN.R-project.org/package=plotly

Wickham, Hadley. 2016. Ggplot2: Elegant Graphics for Data Analysis. Springer-Verlag New York. https://ggplot2.tidyverse.org

Wickham, Hadley, and Jennifer Bryan. 2022. Readxl: Read Excel Files. https://CRAN.R-project.org/package=readxl

Wickham, Hadley, Winston Chang, Lionel Henry, Thomas Lin Pedersen, Kohske Takahashi, Claus Wilke, Kara Woo, Hiroaki Yutani, and Dewey Dunnington. 2022. Ggplot2: Create Elegant Data Visualisations Using the Grammar of Graphics. https://CRAN.R-project.org/package=ggplot2

Wickham, Hadley, Romain François, Lionel Henry, and Kirill Mßller. 2022. Dplyr: A Grammar of Data Manipulation. https://CRAN.R-project.org/package=dplyr

Zeileis, Achim, and Gabor Grothendieck. 2005. “Zoo: S3 Infrastructure for Regular and Irregular Time Series.” Journal of Statistical Software 14 (6): 1–27. https://doi.org/10.18637/jss.v014.i06

Zeileis, Achim, Gabor Grothendieck, and Jeffrey A. Ryan. 2022. Zoo: S3 Infrastructure for Regular and Irregular Time Series (z’s Ordered Observations). https://zoo.R-Forge.R-project.org/

Zeileis, Achim, and Torsten Hothorn. 2002. “Diagnostic Checking in Regression Relationships.” R News 2 (3): 7–10. https://CRAN.R-project.org/doc/Rnews/

Zhu, Hao. 2021. kableExtra: Construct Complex Table with Kable and Pipe Syntax. https://CRAN.R-project.org/package=kableExtra